Mathematics for the interested outsider

Submodules

Fancy words: a submodule is a subobject in the category of group representations. What this means is that if and are -modules, and if we have an injective morphism of modules , then we say that is a “submodule” of . And, just to be clear, a -morphism is injective if and only if it’s injective as a linear map; its kernel is zero. We call the “inclusion map” of the submodule.

In practice, we often identify a -submodule with the image of its inclusion map. We know from general principles that since is injective, then is isomorphic to its image, so this isn’t really a big difference. What we can tell, though, is that the action of sends the image back into itself.

That is, let’s say that is the image of some vector . I say that for any group element , acting by on gives us some other vector that’s also in the image of . Indeed, we check that

which is again in the image of , as asserted. We say that the image of is “-invariant”.

The flip side of this is that any time we find such a -invariant subspace of , it gives us a submodule. That is, if is a -module, and is a -invariant subspace, then we can define a new representation on by restriction: . The inclusion map that takes any vector and considers it as a vector in clearly intertwines the original action and the restricted action , and its kernel is trivial. Thus constitutes a -submodule.

That is, consists of all vectors for which all the coefficients are equal. I say that this subspace is -invariant. Indeed, we calculate

But this last sum runs through all the elements of , just in a different order. That is, , and so carries the one-dimensional trivial representation of . That is, we’ve found a copy of the trivial representation of as a submodule of the left regular representation.

As another example, let be one of the symmetric groups. Again, let carry the left regular representation, but now let be the one-dimensional space spanned by

It’s a straightforward exercise to show that is a one-dimensional submodule carrying a copy of the signum representation.

Every -module contains two obvious submodules: the zero subspace and the entire space itself are both clearly -invariant. We call these submodules “trivial”, and all others “nontrivial”.

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.